For requesting, clarifying, and comparing definitions of mathematical terms.

learn more… | top users | synonyms

1
vote
0answers
34 views

Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: ...
1
vote
0answers
41 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
1
vote
0answers
32 views

Difficulty with terminology/standard definition for Jordan-normal form.

(Perhaps this is something rather for MOverflow, don't know) I've understood the concept of the Jordan-normal-form such that it is similar to the idea of diagonalization of a matrix, but can be ...
1
vote
0answers
91 views

Algebraic definition or construction of real numbers

Is there any algebraic definition or construction of real numbers ? If not, why ?
1
vote
0answers
56 views

Definition: foci of a quadric

How are the foci of a quadric defined? By a quadric I mean a set $$ Q = \left\{ x \in \mathbb R^n \mid x^T A x + 2 b^T x + c = 0 \right\}, $$ where $A\in\mathbb R^{n\times n}$ is symmetric and ...
1
vote
0answers
37 views

What is $(j,\epsilon)$-normality?

In looking at the concept of normality for real numbers I have come across the notion of $(j,\epsilon)$-normality, but cannot find a definition for this. could anyone explain what this term means?
1
vote
0answers
35 views

In a translation of a classic math book: is the term *normal* translated correctly? Is there a better term?

I am reading an English translation of the classic book by Gelfond & Linnik: Elementary Methods in the Analytic Theory of Numbers. Here is the definition of "normal" from the book (page 5 in my ...
1
vote
0answers
68 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
1
vote
0answers
19 views

How do I state a reduction in cost?

I have developed an algorithm and am having a hard time stating its benefit versus a baseline. Suppose that the baseline cost of solving the problem is 1000 seconds. Now suppose that my algorithm ...
1
vote
0answers
85 views

$ |\sin (x) | \leq 1$ by it's definition

I was told that it can be shown that $|\sin(x)| \leq 1$ by its definition $\sin (z):= \frac{1}{2i} \bigl( (\exp(iz)-\exp(-iz)\bigr) $ I am aware that as soon as I choose $x \in \mathbb{R}$ and ...
1
vote
0answers
43 views

Subgraph without “holes”

does everyone know, if there already exists a definition of subgraphs, which do not contain a "hole"? EDITED: That means: I presuppose a planar embedding of a graph G and I want to find a connected ...
1
vote
0answers
24 views

What's the Denominator of the Fraction in the wake of a Ratio Called?

Say some elixir exists in the ratio : $x_1$ of ingredient 1 $\; \LARGE{:} \;$ $x_2$ of ingredient 2 $\; \LARGE{:} \;$ ... $\; \LARGE{:} \;$ $x_n$ of ingredient n Then the ...
1
vote
0answers
89 views

$f: E \to \mathbb{R}^m$ is not continuous at $x_0$, show that $g:E \cup \lbrace x_0 \rbrace \to \mathbb{R}^m$ is continuous at $x_0$

Let $E \subset \mathbb{R}^k, f: E \to \mathbb{R^m} $ be a function. Let $x_0 \in \mathbb{R}^k$ such that $x_0 \notin E$ but note that $ \displaystyle \lim_{x \to x_0}_{x \in E}=y_0$ does exist. ...
1
vote
0answers
153 views

Difference between “marginal variance” and “true variance”?

[This question is not about solving a mathematical problem, but about definitions; please tell me if there is a better forum to ask it.] In a paper I am reading, the author writes: ...the sole ...
1
vote
0answers
79 views

About definition of topology

let be $m$ a function "$m: X \to \mathcal{P}(\mathcal{P}(X))$" (and I denote: $m(x) := m_x, \forall x \in X $), $m$ is topology on $X$ if: 2)$ \forall x \in X (\forall t \in m_x(x \in t)) $ 3)$ ...
1
vote
0answers
55 views

evaluating a meromorphic section of a line bundle at a point

Let $D$ be a Cartier divisor of a variety $X/K$ with associated line bundle $\mathcal{O}(D)$ and meromorphic section $s_D$. How do you define $s_D(P) \in K$ for $P \in X(K) \setminus ...
1
vote
0answers
350 views

Arc Length Parametrization

Professor was a little fuzzy on this topic, so I just wanted to make sure I have this definition correct: Given a function $\alpha : T \to C \mid t \in [a,b]$ , where $t$ is the parameter and $C$ is ...
1
vote
0answers
42 views

What's a algebraically isolated singularirty?

What's its formal definition and how does it differ from a "simple" isolated singularity? I'm thinking about integrable $1$-forms here, though the definition may be more general.
1
vote
0answers
66 views

How to Define an Infinite Anti-Diagonal Matrix

We can define an infinite diagonal matrix with some ease, and then say that a finite diagonal matrix is the top left sub-matrix of our infinite one. Can we define an infinite anti-diagonal matrix? It ...
1
vote
0answers
55 views

Is a single nontrivial convex set in a topological vector space enough to make it locally convex?

This is sort of a definition question. While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of ...
1
vote
0answers
60 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
1
vote
0answers
88 views

Which cut does the “minimum cut” refer to?

My course notes give the following definitions; could someone please verify that the last definition is non-standard? (I've spent all evening googling, and isn't "minimum cut" a concept related to cut ...
1
vote
0answers
112 views

How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then ...
1
vote
0answers
207 views

Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
1
vote
0answers
37 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
1
vote
0answers
301 views

different kind of convergence in Real analysis

Now I am studying real analysis, I wonder if anyone can sum up all the relations between all kinds of convergence, convergence in measure, convergence a.e., convergence a.u., and convergence in ...
1
vote
0answers
84 views

Defining things in a non-recursive manner

In this question I asked, if it was possible to define certain functions without the use of the recursion theorem. The answer, among other things, indicated that it theoretically would be possible, ...
1
vote
0answers
24 views

Definition of Chromatic class digraph of a m-coloured digraph

I have to prove that if $D$ is a m-coloured digraph, then $K(D) = K\big(C(D)\big)$ Where $C(D)$ is the chromatic transitive closure of $D$, which is a multidigraph with the same nodes and arcs of $D$ ...
1
vote
0answers
156 views

How to prove the definition of arctangent by G. H. Hardy through integral?

From introduction to analysis,by Arthur P. Mattuck,problem 20-1. I am stuck in the sub-problem (d) of this problem,especially the magic number 2.5,please help,thanks. Problems 20-1 One way of ...
1
vote
0answers
98 views

How are the various numbers in the standard 2.2 gamma correction for RGB derived?

Here is the standard fwd Gamma 2.22 (1 / 0.45) correction formula: ...
1
vote
0answers
69 views

Definition of fragility

What does it mean for a solution to a system of differential equations to be fragile? A context for the term can be found here: This is taken from here in Mathematical Methods for Mechanics: A ...
1
vote
0answers
61 views

Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same ...
1
vote
0answers
95 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
1
vote
0answers
173 views

Is the extra condition in this definition superfluous?

I am learning Differential Geometry and someone told me that the second condition of a definition provided in books follows from the first and is hence superfluous. I cannot dispute it, so convince me ...
0
votes
0answers
17 views

Definition of a vector field

Reading Wikipedia, I see that a vector field is defined as a mapping $X: S \rightarrow \mathbb R^n$ where $S \subseteq R^n$. However, I sometimes see mappings $X: S \subseteq R^m \rightarrow R^n$ ...
0
votes
0answers
11 views

What is the correct term for the equations which comprise a mathematical model?

I have a mathematical model constructed by myself and my supervisor. In writing my report do I refer to the equations that make up this model as "constitutive equations", or is there another term ...
0
votes
0answers
17 views

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$ (or $n-1$ and proving for $n$)? both with induction. The first one is ...
0
votes
0answers
7 views

$k$-ary labeled trees with distinct labels

Classical definition of $k$-ary labeled trees doesn't restrict somehow the uniqueness of tree labels inside its branches. My question: Is any special definition (name) for such trees? To clarify ...
0
votes
0answers
23 views

Holomorphs and split extensions

The notion holomorph was introduced in Maria S. Voloshina's Ph.D. thesis On the Holomorph of a Discrete Group. It is defined as follows: Let $G$ be a group and let $\mathrm{Aut}(G)$ be the ...
0
votes
0answers
27 views

Difference between upper bound, maximal element, and maximum

In order to learn the difference between upper bound, maximum, and maximal element (of a set), I wrote down the following. Is it correct? Upper bound not necessarily element (of set) greater than ...
0
votes
0answers
15 views

What is meant by “commutes with spatial translations”?

Let $f\colon(X,\mathcal{B},m_1)\to(X,\mathcal{B},m_2)$, where $(X,\mathcal{B},m_i), i=1,2$ are measure spaces. What is meant when saying that f commutes with all spatial translations?
0
votes
0answers
14 views

$k$-cube definition clarrification

If $c:I^k\rightarrow\mathbb{R}^n$ is a singular $k$-cube. What will $\partial c$ be? Will it be a singular $(k-1)$-cube or a $(k+1)$-cube or something else? Definition - A singular $k$-cube on $U$ ...
0
votes
0answers
37 views

Is there a name for a function that returns only binary values?

Is there a name for a function that returns only binary values (e.g., $f(x) : X \rightarrow \{0,1\}$)?
0
votes
0answers
6 views

What does bimeasurable mean? Is an invertible transformation bimeasurable?

What exactly is the meaning of a bimeasurable transformation? I did not find a very clear answer to that. As far as I see it means that Borelsets are maped to Borelsets. So an invertible ...
0
votes
0answers
13 views

Word for two objects with coplanar axes.

Is there a suitable word for describing two objects with coplanar axes (e.g. cylinders)? The word parallel springs to mind, but I wondered if there was anything more specific.
0
votes
0answers
38 views

About a variation of the primitive root idea.

Let $n $ be called a half primitive root mod ($m$) if $n^{\phi(m)/2} = 1$ mod($m$) and for any $t $ with $1 < t < \phi(m)/2$, $m$ does not divide $(n^t - 1)$. So the order of $n$ mod($m$) is ...
0
votes
0answers
31 views

Well defined uncomputable numbers.

For any prefix-free universal computable function $F$ with domain $P_F$, the Chaitin’s constant $$ \Omega_F=\sum_{p\in P_F}2^{-|p|} $$ is a number $\in [0,1]$ and seems "well defined". But this ...
0
votes
0answers
28 views

Set of functions and sequences

By $A^G$ I mean $\left\{x\colon G\to A\right\}$. Is it then to same to write $$ A^G=\left\{x=(x_g)_{g\in G}, x_g\in A\right\}? $$
0
votes
0answers
48 views

What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
0
votes
0answers
9 views

Definition of Bernouilli shift

Let $\mathfrak{B}$ denote the Borel field on $X$ generated by its topology and let $\mu_{p_0,p_1,p_2}$ be product measure on $X$ in which the $i$'s have density $p_i$. A translation-invariant ...